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DOE OSTI2017

Stability analysis of nonlinear two-grid method for multigroup neutron diffusion problems

We present theoretical analysis of a nonlinear acceleration method for solving multigroup neutron diffusion problems. This method is formulated with two energy grids that are defined by (i) fine-energy groups structure and (ii) coarse grid with just a single energy group. The coarse-grid equations are derived by averaging of the multigroup diffusion equations over energy. The method uses a nonlinear prolongation operator. We perform stability analysis of iteration algorithms for inhomogeneous (fixed-source) and eigenvalue neutron diffusion problems. To apply Fourier analysis the equations of the method are linearized about solutions of infinite-medium problems. The developed analysis enables us to predict convergence properties of this two-grid method in different types of problems. Numerical results of problems in 2D Cartesian geometry are presented to confirm theoretical predictions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2012

Introduction of corrections taking into account interdependence of multigroup constants to the results of multigroup perturbation theory calculations

In multigroup calculations of reactivity and sensitivity coefficients, methodical errors can appear if the interdependence of multigroup constants is not taken into account. For this effect to be taken into account, so-called implicit components of the aforementioned values are introduced. A simple technique for computing these values is proposed. It is based on the use of subgroup parameters.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2019

The SCALE/AMPX multigroup cross section processing for fast reactor analysis

The SCALE/AMPX multigroup (MG) cross section processing procedure has been updated to minimize reactivity differences for fast reactor designs and boiling water reactors (BWRs) with very high void fractions to provide excellent agreement with continuous-energy reference calculations. SCALE MG calculations are widely applied to thermal spectrum light-water reactor (LWR) systems as well as fast spectrum metallic systems of interest to National Nuclear Security Administration. With growing interest from industry and regulators in applying SCALE for the design of fast spectrum reactors, both sodium and molten salt, and in the licensing of power up rates for BWRs, it is desirable to review the SCALE/AMPX procedure for unresolved resonance self-shielded data and high energy neutron spectra. The data were improved by generating MG unresolved resonance data based on the analytic probability table method with the narrow resonance approximation as well as the use of a very fine group structure typically used in fast system analysis. This study focused on verifying the probability table and the SCALE/AMPX MG cross section processing procedure by performing reaction rate analysis and benchmark calculations for various fast reactor and high void BWR systems. Here, results indicate that the improved SCALE/AMPX MG cross section processing provides excellent results for the fast reactor analysis.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2021

The Effect of the Flux Separability Approximation on Multigroup Neutron Transport

The angular dependence of flux-weighted multigroup cross sections is commonly neglected when generating multigroup libraries. The error of this flux separability approximation is typically not isolated from other error sources due to a lack of availability of library generation and corresponding solvers that cannot relax this approximation. These errors can now be isolated and quantified with the availability of a multigroup Monte Carlo transport and multigroup library-generation capability in the OpenMC Monte Carlo transport code. This work will discuss relevant details of the OpenMC implementation, provide an example case useful for detailing the type of errors one can expect from making the flux separability approximation, and end with more realistic problems which show the impact of the approximation and highlight how it can strongly arise from an energy-dependent resonance absorption effect. Since the angle-dependence is intrinsically linked to the energy group structure, these examples also show that relaxing the flux separability approximation with angle-dependent cross sections could be used to reduce either the fine-tuning required to set a multigroup energy structure for a specific reactor type or the number of energy groups required to obtain a desired level of accuracy for a given problem. This trade-off could increase the costs of generating multigroup cross sections, and has the potential to require more memory for storing the multigroup library during the transport calculations, but it can significantly reduce the computational time required since the runtime of a discrete ordinates or method of characteristics neutron transport solver scales roughly linearly with the number of groups.

42 ENGINEERING↗

DOE OSTI2020

Multigroup Constant Calculation with Static α-Eigenvalue Monte Carlo for Time-Dependent Neutron Transport Simulations

Herein, multigroup constants for deterministic methods that preserve the time-dependent physics of the neutron transport equations are derived. Alternative multigroup constant weighting spectra are discussed: (1) the fundamental k-eigenfunction, (2) the fundamental α-eigenfunction, and (3) a composite of several α-modes. To generate the fundamental α-eigenfunction for calculating the multigroup constants, a static fundamental α-eigenvalue method is implemented into the open source Monte Carlo code OpenMC. Several static and kinetic problems are devised to verify the implementations and to investigate the relative performance of the alternative multigroup constant weighting spectra. Results emphasize that as a multigroup constant weighting spectrum, the fundamental α-eigenfunction offers physical characteristics that make it advantageous (in producing accurate solutions) over the typically used fundamental k-eigenfunction.

42 ENGINEERING↗

DOE OSTI2013

MC 2 -3: Multigroup Cross Section Generation Code for Fast Reactor Analysis

The MC 2 -3 code is a Multigroup Cross section generation Code for fast reactor analysis, developed by improving the resonance self-shielding and spectrum calculation methods of MC 2 -2 and integrating the one-dimensional cell calculation capabilities of SDX. The code solves the consistent P1 multigroup transport equation using basic neutron data from ENDF/B data files to determine the fundamental mode spectra for use in generating multigroup neutron cross sections. A homogeneous medium or a heterogeneous slab or cylindrical unit cell problem is solved in ultrafine (~2000) or hyperfine (~400,000) group levels. In the resolved resonance range, pointwise cross sections are reconstructed with Doppler broadening at specified isotopic temperatures. The pointwise cross sections are directly used in the hyperfine group calculation whereas for the ultrafine group calculation, self-shielded cross sections are prepared by numerical integration of the pointwise cross sections based upon the narrow resonance approximation. For both the hyperfine and ultrafine group calculations, unresolved resonances are self-shielded using the analytic resonance integral method. The ultrafine group calculation can also be performed for two-dimensional whole-core problems to generate region-dependent broad-group cross sections. Multigroup cross sections are written in the ISOTXS format for a user-specified group structure. The code is executable on UNIX, Linux, and PC Windows systems, and its library includes all isotopes of the ENDF/BVII. 0 data.

20 FOSSIL-FUELED POWER PLANTS↗

DOE OSTI2019

Multigroup Neutron Cross Section Generation for the SCEPTRE Code

Multigroup neutron cross sections were generated for the deterministic radiation transport code SCEPTRE. ENDF/B-VII.1 nuclear data files were downloaded from Los Alamos National Laboratory (LANL), processed with the LANL cross-section preparation code NJOY-2012, and post-processed to produce a SCEPTRE-formatted cross section file. A simple radiation transport problem was used to compare results calculated with MCNP, a continuous-energy radiation transport code from LANL, to results calculated with SCEPTRE using the NJOY-2012-produced multigroup cross sections. This problem was used to debug the python scripts used to post-process the NJOY-2012 output and to assess the accuracy of the multigroup cross sections. These comparisons demonstrate that the multigroup cross sections generated in this work are accurate for most elements but yielded inaccurate results for several common transition metals. This discrepancy appears to result from poor treatment of the resolved resonance region of the continuous-energy cross sections. Further work is recommended to investigate alternative methods to treat these resonances with NJOY-2012.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2016

Multigroup Transport Equations Derived via Homogeneity and Isotropy Restoration Theory

The multigroup neutron transport calculation at a cell-level or assembly-level is a key step in conventional deterministic reactor analysis. As the computing power grows, it is contemplated that the multigroup transport equation itself be used in whole-core calculation. While the continuous-energy transport equation is derived straightforwardly and rigorously, the multigroup transport equation currently in use involves several approximations during its derivation. Recently, we proposed the Homogeneity and Isotropy Restoration (HIRE) theory in a preliminary form for deriving a multigroup transport equation that maintains angle-independence in the total cross section and homogeneity in the material region (originally each material region is usually homogeneous by design). As a byproduct, only the isotropic scattering term remained, which also relieved the computational burden significantly. In the Homogeneity and Isotropy Restoration (HIRE) theory, we introduced a discontinuity factor in the form of partial current discontinuity factor (PCDF) at the material interface. In this paper, we elaborate on the HIRE theory and apply it to several typical pin-cell problems. (authors)

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2018

A fully coupled two-level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations

The neutron multigroup diffusion equations is an important approximation of the neutron transport equation, and has been widely used for studying the motion of neutrons and their interactions with the stationary background materials. Solving the multigroup diffusion equations faces challenging because many variables are coupled together and a lot of computing resources are required. In this paper, we focus on the development of a scalable parallel algorithm framework for the system of equations arising from the discretization of the multigroup diffusion equations using a finite element method in space and using an implicit scheme in time. The parallel agorithm framweork consists of an inexact Jacobian-free Newton for the nonlinear equations, a Krylov subspace method for the Jacobian system and a parallel Schwarz preconditioner for the acceleration of the algorithm convergence. A coarse space based on smoothed aggregation is introduced to construct a two-level method to improve the parallel performance of the Schwarz preconditioner. We numerically shown that compared with the one-level method, the proposed two-level method works significantly better in terms of the number of linear iterations and the compute time for a system of equations with millions of unknowns and a supercomputer with thousands of processors.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Multigroup Cross-Section Generation with the OpenMC Monte Carlo Particle Transport Code

High-fidelity deterministic transport codes require highly accurate multigroup cross sections (MGXS). Monte Carlo is increasingly cited as a reactor-agnostic approach to MGXS generation since it is unconstrained by the engineering-based approximations that limit the applicability of deterministic MGXS generation tools. This paper introduces a new framework that uses the OpenMC Monte Carlo code to generate MGXS for use in multigroup transport codes. The openmc.mgxs module is built atop OpenMC's Python application programming interface to process tally data output by the OpenMC executable. This paper validates the module to generate MGXS that enable the multigroup OpenMOC transport code to compute eigenvalues to within 50 pcm and fission rates to within 1% of reference solutions for two heterogeneous pressurized water reactor benchmarks.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2017

MC 2 -3: Multigroup Cross Section Generation Code for Fast Reactor Analysis

This paper presents the methods and performance of the MC2 -3 code, which is a multigroup cross-section generation code for fast reactor analysis, developed to improve the resonance self-shielding and spectrum calculation methods of MC2 -2 and to simplify the current multistep schemes generating region-dependent broad-group cross sections. Using the basic neutron data from ENDF/B data files, MC2 -3 solves the consistent P1 multigroup transport equation to determine the fundamental mode spectra for use in generating multigroup neutron cross sections. A homogeneous medium or a heterogeneous slab or cylindrical unit cell problem is solved in ultrafine (2082) or hyperfine (~400 000) group levels. In the resolved resonance range, pointwise cross sections are reconstructed with Doppler broadening at specified temperatures. The pointwise cross sections are directly used in the hyperfine group calculation, whereas for the ultrafine group calculation, self-shielded cross sections are prepared by numerical integration of the pointwise cross sections based upon the narrow resonance approximation. For both the hyperfine and ultrafine group calculations, unresolved resonances are self-shielded using the analytic resonance integral method. The ultrafine group calculation can also be performed for a two-dimensional whole-core problem to generate region-dependent broad-group cross sections. Verification tests have been performed using the benchmark problems for various fast critical experiments including Los Alamos National Laboratory critical assemblies; Zero-Power Reactor, Zero-Power Physics Reactor, and Bundesamt für Strahlenschutz experiments; Monju start-up core; and Advanced Burner Test Reactor. Verification and validation results with ENDF/B-VII.0 data indicated that eigenvalues from MC2 -3/DIF3D agreed well with Monte Carlo N-Particle5 MCNP5 or VIM Monte Carlo solutions within 200 pcm and regionwise one-group fluxes were in good agreement with Monte Carlo solutions.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2023

Deep Learning for Multigroup Cross-Section Representation in Two-Step Core Calculations

Here we investigate using deep learning, a type of machine-learning algorithm employing multiple layers of artificial neurons, for the mathematical representation of multigroup cross sections for use in the Griffin reactor multiphysics code for two-step deterministic neutronics calculations. A three-dimensional fuel element typical of a high-temperature gas reactor as well as a two-dimensional sodium-cooled fast reactor lattice are modeled using the Serpent Monte Carlo code, and multigroup macroscopic cross sections are generated for various state parameters to produce a training data set and a separate validation data set. A fully connected, feedforward neural network is trained using the open-source PyTorch machine-learning framework, and its accuracy is compared against the standard piecewise linear interpolation model. Additionally, we provide in this work a generic technique for propagating the cross-section model errors up to the k eff using sensitivity coefficients with the first-order uncertainty propagation rule. Quantifying the eigenvalue error due to the cross-section regression errors is especially practical for appropriately selecting the mathematical representation of the cross sections. We demonstrate that the artificial neural network model produces lower errors and therefore enables better accuracy relative to the piecewise linear model when the cross sections exhibit nonlinear dependencies; especially when a coarse grid is employed, where the errors can be halved by the artificial neural network. However, for linearly dependent multigroup cross sections as found for the sodium-cooled fast reactor case, a simpler linear regression outperforms deeper networks.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2013

The Suppression of Energy Discretization Errors in Multigroup Transport Calculations

The Objective of this project is to develop, implement, and test new deterministric methods to solve, as efficiently as possible, multigroup neutron transport problems having an extremely large number of groups. Our approach was to (i) use the standard CMFD method to "coarsen" the space-angle grid, yielding a multigroup diffusion equation, and (ii) use a new multigrid-in-space-and-energy technique to efficiently solve the multigroup diffusion problem. The overall strategy of (i) how to coarsen the spatial an energy grids, and (ii) how to navigate through the various grids, has the goal of minimizing the overall computational effort. This approach yields not only the fine-grid solution, but also coarse-group flux-weighted cross sections that can be used for other related problems.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2015

Multigroup Free-atom Doppler-broadening Approximation. Theory

Multigroup cross sections at a one target temperature can be Doppler-broadened to multigroup cross sections at a higher target temperature by matrix multiplication if the group structure suf- ficiently resolves the original temperature continuous energy cross section. Matrix elements are the higher temperature group weighted averages of the integral over the lower temperature group boundaries of the free-atom Doppler-broadening kernel. The results match theory for constant and 1/v multigroup cross sections at 618 lanl group structure resolution.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2013

CASTRO: A NEW COMPRESSIBLE ASTROPHYSICAL SOLVER. III. MULTIGROUP RADIATION HYDRODYNAMICS

We present a formulation for multigroup radiation hydrodynamics that is correct to order O(v/c) using the comoving-frame approach and the flux-limited diffusion approximation. We describe a numerical algorithm for solving the system, implemented in the compressible astrophysics code, CASTRO. CASTRO uses a Eulerian grid with block-structured adaptive mesh refinement based on a nested hierarchy of logically rectangular variable-sized grids with simultaneous refinement in both space and time. In our multigroup radiation solver, the system is split into three parts: one part that couples the radiation and fluid in a hyperbolic subsystem, another part that advects the radiation in frequency space, and a parabolic part that evolves radiation diffusion and source-sink terms. The hyperbolic subsystem and the frequency space advection are solved explicitly with high-order Godunov schemes, whereas the parabolic part is solved implicitly with a first-order backward Euler method. Our multigroup radiation solver works for both neutrino and photon radiation.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2018

MC 2 -3: Multigroup Cross Section Generation Code for Fast Reactor Analysis Nuclear

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2020

Release of Five Multigroup Neutron Libraries: e68g_103, mendf80, mt80, mendf80_30, and mt80_30

Five multigroup libraries are discussed for external release. They are e68g_103, a multigroup photon library, and mendf80, mt80, mendf80_30 and mt80_30 are multigroup neutron libraries.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Multigroup Cross-section Generation in MCNP6.3 [Slides]

This presentation states that in comparison to the NJOY-produced multigroup cross sections, the MCNP-produced multigroup cross sections are generally consistent. Statistical uncertainties, however, are challenging and the unresolved resonance region may be looked at in the future. It also discusses how the SPM and LCS options were compared to each other for internal consistency. Additionally, some reactor pin-cell-like problems were used to compare to multigroup capabilities in other Monte Carlo codes (e.g., Serpent, OpenMC).

22 GENERAL STUDIES OF NUCLEAR REACTORS↗